Transcript LINEAR RELATIONSHIPS

Representing Linear
Non-proportional Relationships
Essential Question?
How can you use tables, graphs,
mapping diagrams, and equations to
represent linear non-proportional
situations?
8.F.3
Common Core Standard:
8.F.3 ─ Define, evaluate, and compare functions.
Interpret the equation y = mx + b as defining a linear function, whose
graph is a straight line; give examples of functions that are not linear.
For example, the function A = s2 giving the area of a square as a
function of its side length is not linear because its graph contains the
points (1,1), (2,4) and (3,9), which are not on a straight line.
Objectives:
• Understand that a linear function has a constant rate of
change called slope.
• Identify whether a relationship is a linear function from a
diagram, table of values, graph, or equation.
Curriculum Vocabulary
Linear Equation (ecuación lineal):
An equation whose solutions form a straight line on a
coordinate plane.
Slope Intercept Form (forma de pendiente-intersección):
A linear equation written in the form y = mx+b, where
m represents the slope and b represents the y-intercept.
y-intercept (intersección con el eje y):
The y-coordinate of the point where the graph of a function
crosses the y-axis.
LINEAR RELATIONSHIPS
Let’s examine the following situation:
Hertz rental car charges $35 per day plus $0.50 per mile to
rent one of their pick-up trucks. Create a table, write an
equation, and draw a graph for this situation.
Number of miles driven (x)
0
Cost in dollars (y)
What does it cost to drive 0 miles (initial value )?
What other numbers would you choose for the table?
LINEAR RELATIONSHIPS
Hertz rental car charges $35 per day plus $0.50 per mile to rent
one of their pick-up trucks. Create a table, write an equation,
and draw a graph for this situation.
Number of miles driven (x)
0
5
10
15
20
Cost in dollars (y)
35
37.50
40
42.50
45
Do I have a Function?
YES!
Do I have a Constant Rate of Change?
YES!
What is my Constant Rate of Change?
𝟓
𝟏
= = 𝟎. 𝟓𝟎
𝟏𝟎 𝟐
What is my starting point (initial value)?
(0,35)
LINEAR RELATIONSHIPS
A LINEAR RELATIONSHIP can be described by
an equation in the form 𝒚 = 𝒎𝒙 + 𝒃, where 𝒎 is the
SLOPE and b is the y-INTERCEPT.
The y-intercept is the same as the initial value and
should ALWAYS be written as the ordered pair (0,b)
even though many books don’t show it that way.
LINEAR RELATIONSHIPS
Let’s continue with the Hertz rental car.
Number of miles driven (x)
0
5
10
15
20
Cost in dollars (y)
35
37.50
40
42.50
45
What is the slope?
What is the y-intercept?
What is the linear equation?
𝒎=
𝟓
𝟏
= = 𝟎. 𝟓
𝟏𝟎 𝟐
(0,35) 𝒃 = 𝟑𝟓
𝒚=
𝟏
𝒙 + 𝟑𝟓
𝟐
or
𝒚 = 𝟎. 𝟓𝒙 + 𝟑𝟓
LINEAR RELATIONSHIPS
Now let’s graph the function.
What would be the best
scale for my graph?
Number
of miles
driven (x)
Cost in
dollars (y)
0
35
5
37.5
10
40
15
42.5
20
45
LINEAR RELATIONSHIPS
We have now represented the linear relationship as a table,
an equation, and a graph.
Number
of miles
driven (x)
50
Cost in
dollars (y)
45
40
35
5
37.5
10
40
15
42.5
35
Cost in dollars
0
30
25
20
15
10
20
45
1
𝑦 = 𝑥 + 35
2
5
0
5
10
15
20
Number of miles driven
LINEAR RELATIONSHIPS
Now let’s consider the following situation:
José went to the Riverside County Fair & Date Palm Festival
in Indio. Cost of admission was $20. He also decided to go
on some rides. Ride tickets cost $2 each. Create a table,
equation, and graph to represent this scenario.
Number of tickets (x)
0
Price paid in dollars (y)
What does it cost José to enter the festival, even if he
doesn’t purchase any ride tickets (initial value )?
What other numbers would you choose for the table?
LINEAR RELATIONSHIPS
José went to the Riverside County Fair & Date Palm Festival in
Indio. Cost of admission was $20. He also decided to go on
some rides. Ride tickets cost $2 each. Create a table, equation,
and graph to represent this scenario.
Number of tickets (x)
0
1
2
3
4
Price paid in dollars (y)
20
22
24
26
28
Do I have a Function?
YES!
Do I have a Constant Rate of Change?
YES!
What is my Constant Rate of Change?
𝟐
=𝟐
𝟏
What is my starting point (initial value)?
(0,20)
LINEAR RELATIONSHIPS
Let’s continue:
Number of tickets (x)
0
1
2
3
4
Price paid in dollars (y)
20
22
24
26
28
𝟐
=𝟐
𝟏
What is the slope?
𝒎=
What is the y-intercept?
(0,20) 𝒃 = 𝟐𝟎
What is the linear equation?
𝒚 = 𝟐𝒙 + 𝟐𝟎
LINEAR RELATIONSHIPS
Now let’s graph the function.
What would be the best
scale for my graph?
Number
of tickets
(x)
Price paid
in dollars
(y)
0
20
1
22
2
24
3
26
4
28
Would it make sense to connect these points
with a line? Why/Why not?
LINEAR RELATIONSHIPS
We have now represented the linear relationship as a table,
an equation, and a graph.
Number
of tickets
(x)
Price paid
in dollars
(y)
50
45
0
20
1
22
2
24
3
26
4
28
𝑦 = 2𝑥 + 20
Price paid in dollars
40
35
30
25
20
15
10
5
0
1
2
3
4
Number of tickets purchased
Try This One!
Your mom fills the 12 gallon gas
tank in her car with gas. On
average her car gets 37 miles to
the gallon.
Create a table,
equation, and graph showing
how far your mom can drive
before she runs out of gas.